# Introduction to thermodynamics

*This section is meant for those readers, who so far have no basic knowledge of thermodynamics, but who would like to be able to follow the thermodynamic considerations in the following chapters. For further information on thermodynamics we recommend for example [2]*

# Thermodynamic condition of state

The thermodynamic condition of state of a gas is described by 3 parameters. These parameters are linked by a function, so that only 2 parameters (eg pressure and temperature) are necessary to define a condition of state.

Over a large range of conditions this functional link is given with a good approximation by the simple **characteristic equation of an ideal gas :**

Thereby *R* is called the **characteristic gasconstant. ***R*is depending on the type of gas**.**The product of the characteristic gas constant and the relative molecular mass is a constant for all gases.

The thermodynamic behaviour of a gas is dependent on its **isentropic exponent** *kappa*. The value of *kappa* is dependent on the number of atoms in the molecule of the gas.

The deviation of a gas from the ideal gas becomes the greater, the nearer the gas condition is to the liquid state. This is then called **real gas behaviour**.

**Order to understand the working principles of compressors the assumption of ideal gas behaviour is sufficient. We therefore limit ourselves to this behaviour in the following** **chapters**.

# Variations of the conditions of state

Through external influences (on compressors by input of work or extraction of heat) the gas changes its thermodynamic condition of state.

Changes of state without any losses are **reversible** and are described by the **polytropic equation.** The **polytropic exponent** n characterises this change of state.

In order to evaluate such changes, **parameters for** **changes of state for energy and processes **are used. For the evaluation of the the compression of gases the parameters **technical work** and **heat** , also the thermal parameters **enthalpy** and **entropy** are preferentially used.

For each change of state the **law of conservation of energy** (first law of thermodynamics) applies. This law says , that the input of technical work plus heat equals the increase in enthalpy of the gas. When compressing gas and applying cooling, the rise in temperature of the gas is proportional to the difference between the input of work and heat removed.

In order to evaluate energywise changes of state the following **condition of state diagrams** are used :

The **p,v-diagram **showsthe reduction in volume when compressing from condition 1 to condition 2. This reduction is the greater, the smaller the polytropic exponent *n* .The technical work is given as the area below the compression curve and the ordinate.

The **T**,**s-diagram** shows the temperature rise when compressing gas. If the gas is being cooled during compression, then* n* < *kappa* and the entropy reduces.

When heat is added (for example through friction of flow in a turbo compressor rotor ) , then *n > kappa* and the entropy increases. The amount of heat transferred or removed is given by the area under the curve.

# Isothermal and Isentropic compression

These are special cases of the polytropic process and are of particular importance for gas compression :

**Isothermal compression** ( *T = const* ) represents a non achieveable limit of a compression with ideal cooling ( the cooling of the compressor requires a sufficiently high temperature difference to the coolant ).

**Isentropic compression ** (s = *const* ) represents the limit of non cooled compression without internal losses.

The table gives the most important parameters for the evaluation of the two changes of state. It follows from the 1. law of thermodynamics, that the same amount of heat has to be removed as technical work was done. With isentropic compression the rise in enthalpy equals the technical work done.

**The work done with isothermal compression represents the theoretical minimum of required work and is therefore taken as a base value to compare with the actual work done .**

# Multistage compression

The technical work done rises with the compression ratio at isentropic ( noncooled ) and isothermal ( ideally cooled) compression. The rise is the greater, the higher the isentropic exponent.

For safety reasons the **outlet temperature is limited** to a maximum of 135 to 200 deg. C , depending on the type of compressor. This in turn limits the **allowable compression ratio** in single stage compression.

However, multistage compression with interstage cooling is also used to **save energy** .

As an **example** we examine an **ideal two stage compression** ( isentropic compression with total isobaric cooling back to suction temperature ).

The p,v diagram shows the approximation to an isothermal change of state and the reduction in specific work done. The T,s diagram shows the achieved reduction in outlet temperature.